DYNAMIC ANALYSIS OF A UNIFORM CANTILEVER BEAM CARRYING A NUMBER OF ELASTICALLY MOUNTED POINT MASSES WITH DAMPERS

Abstract

The free and forced vibration analyses of a uniform cantilever beam carrying a number of spring–damper–mass systems with arbitrary magnitudes and locations were made by means of the analytical-and-numerical-combined method (ANCM). First of all, a method was presented to replace each “spring–damper–mass” system by a massless equivalent “spring–damper” system with effective spring constant keffand effective damping coefficient Ceffso that the ANCM is available for the title problem. Next, the equation of motion for the “constrained” beam (with spring–damper–mass systems attached) was derived by using the natural frequencies and normal mode shapes of the “unconstrained” beam (without carrying any attachments) incorporating the expansion theorem. Finally, the eigenvalues and the forced vibration responses of the “constrained” beam were determined by conventional numerical methods. To confirm the reliability of the presented theory, all the numerical results obtained from the ANCM were compared with the corresponding ones obtained from the conventional finite element method (FEM) and good agreement was achieved. The influence of the damping magnitude of each spring–damper–mass system on the eigenvalues and the forced vibration responses of the constrained beam was studied. Because the order of the overall property matrices for the equation of motion of the constrained beam derived from the ANCM is much lower than that from the conventional FEM, the storing memory and the CPU time required by the ANCM are much less than those required by the FEM.